Question

Compute maximum value of k that will make the discriminant of the quadratic equation positive \[{x^2} - 6x + k = 0\]

Answer 1

Hint: Here in this question we have to find the value of k, the equation is of the form of the quadratic equation. Here they have mentioned the discriminant is positive. The discriminant is given by \[{b^2} - 4ac\] considering this we determine the value of k and hence we determine the value for the solution.

Complete step by step solution:
The equation is of the form of a quadratic equation. In general the quadratic equation will be in the form of \[a{x^2} + bx + c\]. The discriminant is the part of the quadratic formula underneath the square root symbol: \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions.
If the roots are equal the discriminant is equal to zero. If the roots are positive real the discriminant is greater than zero. If the roots are negative the discriminant is less than zero.
Consider the equation \[{x^2} - 6x + k = 0\]
The discriminant is positive therefore we have \[{b^2} - 4ac > 0\]
Here the value of a is 1 and the value of b is -6 and the value of c is k.
\[ \Rightarrow {( - 6)^2} - 4(1)(k) > 0\]
On simplifying we get
\[ \Rightarrow 36 - 4k > 0\]
Take 4k RHS we get
\[ \Rightarrow 36 > 4k\]
Divide the above equation by 4 we get
\[ \Rightarrow 9 > k\]

Therefore the value of k is less than 9.

Note: The roots depend on the value of discriminant. The discriminant is given by \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions. While simplifying we use the simple arithmetic operations. hence we obtain the required solution.